# Can I make this assumption about the utility function?

My question concerns a household's utility function U(c,l). We're told that the partial derivatives dU/dc and dU/dl > 0, and the second order partial derivatives are negative. Now, can I assume anything about the sign of the partial derivatives Ucl (or Ulc)?

No.

In general, based on the information you have, there are no results you can infer for the partial derivatives $$U_{cl}$$ / $$U_{lc}$$. In this context they could be anything.

As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.

For a "well-behaved" bivariate utility function as regards utiliy maximiaztion, we need the Hessian matrix of partial derivatives to be negative definite, which translates into

$$U_{cc}<0, \;\;\;U_{ll}<0,\;\;\; U_{cc}U_{ll} - \left[U_{cl}\right]^2 >0$$

$$\implies |U_{cl} | < \sqrt{U_{cc}U_{ll}}$$

The first two conditions you have by assumption, the third condition poses a constraint on the relative magnitude in absolute value of the cross-partial, although not on its sign: it has to be smaller in absolute terms than the geometric average of the own-second derivatives, at least at the optimal point.

I notice that in many research papers it is assumed that the cross-partial is zero (utility additive in consumption and leisure).

Nevertheless, this is a good opportunity to apply some economic/behavioral thinking: what would it mean, as regards the relation between consumption and leisure to have $$U_{cl} <0$$? What would it mean to have $$U_{cl} >0$$ ?